In this lab, we will observe Newton's second law using an Atwood machine.^{Hoveroverthese!}

• 1 Pulley w/ String and Stand
• Various Masses
• 1 Photogate setup
  • Photogate
  • Interface box ("LabPro")
  • Computer
• Record data in this Google Sheets data table

An Atwood machine consists of two masses attached to a string draped over a pulley. Ideally, both the string and pulley are massless and the pulley is frictionless.

Adding the forces on each mass gives us the following picture:

Note that, due to our simplifying assumptions, the tension \(F_T\) is the same for each mass (and all the way through the string). We also know \(|\vec{a}_{y1}|=|\vec{a}_{y2}|=a_y\): the velocities are always equal in magnitude (as one moves up, the other moves down), and so the accelerations must be the same also.

As vector equations, the sum-of-forces equations are as follows:

$$m_1\vec{a}_{y1}=\sum \vec{F}_{y1}=\vec{F_T}+\vec{F}_{g1}$$ $$m_2\vec{a}_{y2}=\sum \vec{F}_{y2}=\vec{F_T}+\vec{F}_{g2}$$

If, for each mass' sum-of-forces diagram, we call "positive" the direction which that mass accelerates, then we can reduce these equations to scalar equations. Also plugging in \(F_g=mg\) for each mass, we have:

$$m_1a_y=m_1g-F_T\label{F1}$$ $$m_2a_y=F_T-m_2g\label{F2}$$

A little algebra and we can solve for \(F_T\) and \(a_y\):

$$a_y=\frac{m_1-m_2}{m_1+m_2}g\label{ay}$$ $$F_T=\frac{2m_1m_2}{m_1+m_2}g\label{FT}$$

Begin by opening the "Atwoods" file. Ensure that your pulley is set up with the photogate plugged in. Make sure that when you spin the pulley, the photogate switches between the "blocked" and "unblocked" states.

Click on "Data->User Parameters" and ensure that the photogate distance is entered as .0154m.¹ This is 1/10 the circumference (measured at the radius where the string sits), which is the "distance per spoke" (since the pulley has 10 spokes). The photogate is blocked/unblocked by the spokes, thus this is the distance the masses travel each time the photogate goes through a blocked-unblocked cycle.

If there is not already a string with a loop in each end, make one. Make your range of motion as large as possible: make the length so that when one mass is almost on the floor, the other mass is almost at the pulley. You want at least a half-meter range of motion, and more is better.²

Over the course of this experiment, we will use the following mass combinations:³

• 30g and 50g
• 40g and 60g
• 50g and 70g
• 60g and 80g
• 80g and 100g

Connect the first mass combination.⁴ Hold the string of the light mass in a position where the light mass is low and the heavier mass is high. Keep the string as vertical as possible, to avoid the string falling off the photogate or the masses colliding.⁵

Press the green button, wait a second, then release. Catch the small mass before it hits the pendulum. (If the string falls off the photogate before the end of your trial, retake your data.)

Stop recording. Highlight the region where the data on the velocity vs. time graph looks linear (ignore the beginning and ending parts), and add a linear fit.⁶ If the masses collide, you should only take before that collision; if there is not enough data before the collision to run a good fit, redo the trial.

Record the slope of this fit as the acceleration of the mass for this trial. Then, remove this fit (press the "x") for the next trial.

Perform two more trials with this mass combination.

Finally, repeat (do three trials) for each mass combination.

For each mass combination, compute the average acceleration and the uncertainty in average acceleration using the usual formulas.

Calculate the quantity \(\frac{m_1-m_2}{m_1+m_2}\).⁷ This will not have uncertainty, since we have neglected uncertainty in the masses.

Make a plot of average acceleration vs. \(\frac{m_1-m_2}{m_1+m_2}\) (with appropriate error bars).

Interpret the slope of your graph, and extract from it a measurement of \(g\).

Your TA will ask you to answer some of the following questions (they will tell you which ones to answer):

Experimental questions:

1. If the pulley had a non-negligible mass, what impact would this have on our measurement of \(g\)?
2. If the pulley had non-negligible friction, what impact would this have on our measurement of \(g\)?
3. Suppose you gave the string a short jerk when you released it (either downwards or upwards). What impact would this have on the corresponding measurement of acceleration \(a\)? (Keep in mind: we only measure the acceleration after our initial jerk.)

Theoretical questions:

1. Show that equations \eqref{ay} and \eqref{FT} follow from equations \eqref{F1} and \eqref{F2}. (In other words: do the algebra; solve equations \eqref{F1} and \eqref{F2} for \(F_T\) and \(a_y\).)
2. What would the acceleration and tension be if the two masses were equal?
3. What would the acceleration and tension be if \(m_1\) was much bigger than \(m_2\)? (Hint: In this case, we can make the approximation that \(m_1+m_2\simeq m_1\simeq m_1-m_2\), which allows us to simplify our expressions.)

For further thought:

1. You can treat this system as a single "object." The mass of this object is \(m_1+m_2\) and the total force on this object is the difference of the two gravitational forces, \(F_{g1}-F_{g2}\), because those forces point in "different directions" (one pulling the left mass down and the other pulling the right mass down). Work through the algebra to show that this is mathematically equivalent, insofar as the acceleration is concerned. What assumption(s) allows this scenario to be equivalent?

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

PHY121/122 Plotting Tool